The Hamilton Jacobi Equation For Optimal Control Problems with Discontinuous Time Dependence

Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data

“…Simple examples illustrate that the value function might not be the unique lower semicontinuous generalized (according to the concepts abovementioned) solution to (HJE) in an 'almost everywhere' sense (cf. the discussion in [4]). However, uniqueness properties of the solution can be derived for the mere measurable time dependent case imposing additional conditions on the class of functions which are candidate to be solutions, such as the epigraph of the candidate solution is absolutely continuous w.r.t.…”

“…A different perspective has been recently suggested in [4] for the intermediate case (between the continuous one and the merely measurable one) when the multifunction t F (t, x) has everywhere one-sided limits, for all x, and is continuous on the complement of a zero-measure subset of [S, T ] (without necessarily imposing further a priori regularity conditions such as the absolute continuity of the epigraph of the candidate solutions). In this context, considering optimal control problems with a final cost term (i.e.…”

“…L = 0), the value function turns out to be the unique lower semicontinuous solution to (HJE) taking into account 'everywhere in t' characterizations which involve the concepts of lower Dini derivative and the proximal subdifferential. Further important features of the results obtained in [4] are: the presence of left and right limits F (t + , x) and F (t − , x) (the role of which cannot be exchanged) in the characterizing conditions and the presence of the horizontal proximal subdifferentials in the concept of the proximal solution.…”

“…The main objective of this paper is to explore lower semicontinuous characterizations of the value function in the context of non-autonomous Bolza problems, in which the velocity set F satisfies the same assumptions as in [4]. The Lagrangian L is assumed to have the same behaviour in t (L(•, x, v) is continuous on a set of full measure and has everywhere left and right limits), but is just continuous w.r.t.…”

“…the state variable x, that would be otherwise necessary to impose if we passed to the Mayer form, and which is typically required in previous work for the Mayer problem (cf. [4,12,13,14]). Therefore, the mere state augmentation technique does not simplify the task: we would add a step in the analysis and eventually end up with a (possibly more involved) problem, with exactly the same difficulties as we left the reference optimal control problem in the Bolza form.…”